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Let $A$ be an abelian variety over a number field $K$ and let $\mathcal{A}$ denote its Neron model over $\mathcal{O}_K$. Let $v \in M_K^0$ denote a finite prime of $K$, $k_v$ its residue field, $\mathcal{A}_v = \mathcal{A} \times _{\mathcal{O}_K} k_v$ the special fiber of the reduction of $A$ at $v$, and $\mathcal{A}_v^0$ the connected component of the identity section in the special fiber $\mathcal{A}_v$. Then it is well known that there is a finite group-scheme $\Phi _{A,v}$, s.t. we have an exact sequence of $k_v$-group-schemes

$1 \rightarrow \mathcal{A}_v^0 \rightarrow \mathcal{A}_v \rightarrow \Phi _{A,v} \rightarrow 1$.

Is it true that

$\Phi _{A,v}(k_v) = \mathcal{A}_v(k_v) / \mathcal{A}_v^0(k_v)$?


The motivation for my question is the definition of the Tamagawa number $c_{A,v}$, which in one source is defined as the cardinality of $\Phi _{A,v}(k_v)$ and in another source as the cardinality of the quotient group $\mathcal{A}_v(k_v) / \mathcal{A}_v^0(k_v)$. And a priori one only knows that $\Phi _{A,v}(k_v) = H^0(\mathcal{A}_v(\bar k_v) / \mathcal{A}_v^0(\bar k_v))$.

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Since $\mathcal{A}_v^0$ is connected and $\mathcal{k}_v$ is finite, this follows from Lang's theorem (torsors for connected groups are trivial). – ulrich Jun 17 '11 at 12:14
It is maybe worth pointing out that rational points of component groups of abelian varieties over discrete valuation fields have been investigated by Bosch and Liu in their paper "Rational points of the group of components of a Néron model", Manuscripta Math. 98 (1999), no. 3, 275-293. – Stefano V. Jun 17 '11 at 12:36
Thanks for the answer and the reference! – Stefan Keil Jun 17 '11 at 13:53

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